Figure 7.6The change in gravitational potential energy(ΔPEg)between points A and B is independent of the path.ΔPEg=mghfor any path between the two points.
Gravity is one of a small class of forces where the work done by or against the force depends only on the starting and ending points, not on the path between them.Example 7.6 The Force to Stop Falling
A 60.0-kg person jumps onto the floor from a height of 3.00 m. If he lands stiffly (with his knee joints compressing by 0.500 cm), calculate the
force on the knee joints.
StrategyThis person’s energy is brought to zero in this situation by the work done on him by the floor as he stops. The initialPEgis transformed into
KEas he falls. The work done by the floor reduces this kinetic energy to zero.
Solution
The work done on the person by the floor as he stops is given byW=Fdcosθ= −Fd, (7.29)
with a minus sign because the displacement while stopping and the force from floor are in opposite directions(cosθ= cos 180º = − 1). The
floor removes energy from the system, so it does negative work.The kinetic energy the person has upon reaching the floor is the amount of potential energy lost by falling through heighth:
KE = −ΔPEg= −mgh, (7.30)
The distancedthat the person’s knees bend is much smaller than the heighthof the fall, so the additional change in gravitational potential
energy during the knee bend is ignored.The workW done by the floor on the person stops the person and brings the person’s kinetic energy to zero:
W= −KE =mgh. (7.31)
Combining this equation with the expression forWgives
−Fd=mgh. (7.32)
Recalling thathis negative because the person felldown, the force on the knee joints is given by
(7.33)
F= −
mgh
d
= −
⎛⎝60.0 kg
⎞
⎠⎛⎝9.80 m/s
2 ⎞
⎠(−3.00 m)
5.00×10
−3
m
= 3.53×10^5 N.
Discussion
Such a large force (500 times more than the person’s weight) over the short impact time is enough to break bones. A much better way to cushion
the shock is by bending the legs or rolling on the ground, increasing the time over which the force acts. A bending motion of 0.5 m this way yields232 CHAPTER 7 | WORK, ENERGY, AND ENERGY RESOURCES
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